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it should be evident how A3-C#4 and A#3-D4 have swapped their beat-rates.

Perhaps less evident is the (small) difference (between those two samples), when it comes to C4-E4. If you like, you can exercise your ears ((intonation/flavor) and (beat/speed)) by re-listening and comparing both a few times. At some stage we'll have to be able to manage that.

One more thing... once you refine your middle-octave, expand every time with octaves and check every time 10ths, 12ths and 15ths (going treble) and 4ths, 5ths, 6ths and 10ths (towards the bass). So doing you may better balance your ear-training and the overall straining of the pins.

This time, let's listen to 4ths and 5ths in the (flow-chart) sequence order, we will try to spot over-beating intervals and relate them to the thirds progression.

The base:

A3-A4: Ok A3-E4: Ok, is it inverted? Always get sure...E4-A4: Too close to just, is E4 inverted?A3-D4: Ok, it moves a bit much...D4-A4: Ok, always make sure A4 and D4 are not inverted

A3-E4: Ok...E4-B3: Too just (always compare with A3-D4...)B3-F#4: Ok, right now check your 1st third D4-F#4... it is very tense due to B3 (too high in pitch) F#4-C#4: Too just (always compare with A3-D4 and E4-A4 (*)), check your 2nd third A3-C#4... it is very tense due to C#4 (too high)C#4-G#4: Too much beating, check your third E4-G#4... it is sour, make sure C#4-G#4 is not invertedG#4-D#4: Too much beating (compare E4-A4), check your third B3-D#4 and compare it to A3-C#4... what do you hear? Their progression is inverted, in fact A3-C#4 is much faster.

Here you could go back and check/see what is wrong... In fact, you want B3-D#4 faster than A3-C#4.

Never mind, now consider that all intervals are related to each other, in time you'll be able to mentally draw a "beat-map" and administrate those relationships in the strictest way.

..."I have difficult to here the intonation of fifths."...

Don't worry, you are not alone... bi-chords (only two notes played together/simultaneously) have a fairly large leeway, meaning that our ear may accept, say, F3-A3 at a beat-rate from (anything close to) zero to perhaps 10 bps. It is only when we (and orchestras and chorus) play complex chords that intervals need to be tempered into a (complex) whole.

..."Sometimes I need ETD to confirm if a fifth interval in right side, even I count the beat rate correct."...

I too need to confirm that, and I do that with my tuning hammer: I move it either drop-wise (anti-clock) or raise-wise (clock-wise) and see if the beat gets faster or slower. For example, if a fifth is on the narrow side, ACW will produce some (more) beating, CW will make it still.

The above five intervals represent the basic relation (and reference), even if you need to further correct them subsequently (*).

E4->B3, cannot be slower than A3-D4, not faster than E4-A4B3->F#4, cannot be narrower than A3-E4, not "juster" than D4-A4F#4->C#4, wider than A3-D4, less wide than E4-A4C#4->G#4, very very close to D4-A4, a "hair" narrower G#4->D#4, very very close to E4-A4, a hair slowerD#4->A#3, very very close to A3-D4, a hair widerA#3->F4, very very close to A3-E4, a hair less-narrow

A3-C#4: Fast, beats are in a flowA#3-D4: Slow, about 6 bpsB3-D#4: As aboveC4-E4: Too tenseC#4-F4: Slow, about 4 bpsD4-F#4: Too tenseD#4-F#4: Tense and close to correctE4-G#4: Tense, close to correct F4-A4: Sweet.

Good job indeed. Although thirds are not all precisely progressive, you have managed to make some adjacent thirds very similar, which is good.

Let's listen together.

The base:

A3-A4: Nice, it may be inverted A3-E4: It moves too much...E4-A4: It moves too little. Two main possibilities: first p., E4 is too narrow and A3-A4 is narrow; second p., E4 wide! and A3-A4 correctA3-D4: Ok, it moves a lot (about 2 bps, make it about 1 bps)...D4-A4: It moves too much (almost 3 bps), it should sound almost still. Perhaps (first p.) E4 is too narrow and A3-A4 is narrow

Following the (Flow-chart) sequence order:

A3-E4: It moves too much...E4-B3: Too just (always compare with A3-D4)B3-F#4: It beats too much (about 2 bps, it should beat less than A3-E4, more than D4-A4), right now check your 1st third D4-F#4… it is nice/sweet, it must be nice/tense... a "hair" difference that you will mature in time. Here also check your first 6th A3-F#4, you want to develop the taste.

F#4-C#4: Too just (compare with A3-D4 and E4-A4), check your 2nd third A3-C#4 and compare with D4-F#4… D4-F#4 is sweeter, not in progression, so I would quickly go back and improve previous intervals.

C#4-G#4: Nice, still, check again adjacent D4-A4... very different, I would go back and improve the base;

G#4-D#4: Too much beating, check your third B3-D#4 and compare it to A3-C#4... what do you hear? B3-D#4 is sweeter, their progression is inverted;

..."When hear a fifth have correct wave form, then I stretch the pin clockwise little, without turning the pin to confirm the direction."...

That's good.

..."Fouths seems have larger movement, is easier to confirm direction."...

Ok, when tuning the base, make sure A3-A4 and A3-D4 are wide, make A3-E4 very slowly (narrow) beating and E4-A4 sensibly faster than A3-D4.

..."I also compare triads from A3 up to D4 chromatically. Mistakes are reflected in inconsistent. color."...

I too like doing that and find it useful.

..."Took more than half hour to tune A3-A4. A lot faster than 3 hours, but still too slow in production."...

That's ok, try not to hesitate... Tune your note, compare quickly with other available intervals, do not aim at "static" perfection, get close from above and ready to improve all (improvable) points you could memorize.

Mute the piano and listen again. I can hear the A3-D4 beating to fast indeed. And most intervals I can hear. It same as what I heard during last tuning session. It obviously I don't trust my ear enough.

Will tune again tomorrow and further investigate to A3-A4 to confirm if its too wide or too narrow.

A3-C#4: Fast, beats are in a flowA#3-D4: Slow, about 6 bpsB3-D#4: As aboveC4-E4: Too tenseC#4-F4: Slow, about 4 bpsD4-F#4: Too tenseD#4-G4: Tense and close to correctE4-G#4: Tense, close to correct F4-A4: Sweet.

If you like, you may let me know what you hear. In particular, you may compare and notice how A3-C#4, D#4-G4, E4-G#4 and F4-A4 have changed, thinking about color/tension/taste/texture.

Try to relate beats with the movement of your tuning hammer: in other words, while turning the hammer... do not stop the sound, keep your fingers down onto the keyboard and listen to the interval, how the interval (and the beat-rate) can change.

ThirdsA3-D#4 9 bps, OK (yes, very little slow) A#3-D4 little slower (yes)B3-D#4, 8 bps, slower (I'm not sure, then I would simply check 4ths and 5th on B3 and D#4) C4-E4, seems ok, but not accurate estimation since previous interval is too slow.C#4-F4 same as previous interval, too slow (yes)D4-F#4, D#4-G4, E4-G#4, F4-A4, over impression is too fast, but have progression. (yes, but D#4-G4 is sweeter than the others)

For the time being I have to stop here...

Regards, a.c.

(*)Edit: In general, it is better to stay a "hair" higher, in consideration of the pitch tendency (drop). Many times, you will be able to "hair" correct any interval by playing a Forte. We want to anticipate "hysteresis".